
theorem Th11:
  for F being finite set, A being FinSequence of bool F,
     i being Element of NAT, x being set st i in dom A & x in A.i holds
     card (Cut (A,i,x).i) = card (A.i) - 1
proof
  let F be finite set, A be FinSequence of bool F, i be Element of NAT,
      x be set;
  set f = Cut (A,i,x);
  assume that
A1: i in dom A and
A2: x in A.i;
  i in dom f by A1,Def2;
  then
A3: f.i = A.i \ {x} by Def2;
  {x} c= A.i by A2,ZFMISC_1:31;
  then card (f.i) = card (A.i) - card {x} by A3,CARD_2:44
    .= card (A.i) - 1 by CARD_2:42;
  hence thesis;
end;
